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Lengrand & Miquel (2008). Greco-Roman Fω, orthogonality and symmetrical candidates. Annals of Pure and Put on Logic 153:3-20.

We portray a version of system Fω, bade Fω^C, in which the layer of type
constructors is basically the traditional one of Fω, whereas provability
of types is classic. The proof-term calculus accounting for the classic
reasoning is a variant of Barbanera and Berardi’s symmetric λ-calculus.
We testify that the hale calculus is powerfully normalising. For the
layer of type constructors, we utilize Tait and Girard’s reducibility method
combined with orthogonality techniques. For the (classic) layer of terms,
we expend Barbanera and Berardi’s method based on a symmetrical notion of
reducibility candidate. We try that orthogonality does not catch the
fixpoint construction of symmetrical candidates.

We constitute the consistency of Fω^C, and touch the calculus to the
traditional system Fω, as well when the latter is extended with axioms for
Graeco-Roman logic.